L(s) = 1 | + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)7-s + (0.974 + 0.222i)8-s + (0.781 + 0.623i)10-s + (0.974 − 0.222i)11-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s + i·17-s + (−0.781 − 0.623i)19-s + (0.222 − 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯ |
L(s) = 1 | + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)7-s + (0.974 + 0.222i)8-s + (0.781 + 0.623i)10-s + (0.974 − 0.222i)11-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s + i·17-s + (−0.781 − 0.623i)19-s + (0.222 − 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6955635165 + 0.03476680458i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6955635165 + 0.03476680458i\) |
\(L(1)\) |
\(\approx\) |
\(0.7644587223 - 0.08402910098i\) |
\(L(1)\) |
\(\approx\) |
\(0.7644587223 - 0.08402910098i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.433 - 0.900i)T \) |
| 5 | \( 1 + (-0.900 + 0.433i)T \) |
| 7 | \( 1 + (0.623 + 0.781i)T \) |
| 11 | \( 1 + (0.974 - 0.222i)T \) |
| 13 | \( 1 + (0.222 + 0.974i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (-0.781 - 0.623i)T \) |
| 23 | \( 1 + (0.900 + 0.433i)T \) |
| 31 | \( 1 + (0.433 + 0.900i)T \) |
| 37 | \( 1 + (-0.974 - 0.222i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (-0.433 + 0.900i)T \) |
| 47 | \( 1 + (-0.974 + 0.222i)T \) |
| 53 | \( 1 + (0.900 - 0.433i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (0.781 - 0.623i)T \) |
| 67 | \( 1 + (0.222 - 0.974i)T \) |
| 71 | \( 1 + (-0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.433 - 0.900i)T \) |
| 79 | \( 1 + (0.974 + 0.222i)T \) |
| 83 | \( 1 + (-0.623 + 0.781i)T \) |
| 89 | \( 1 + (-0.433 - 0.900i)T \) |
| 97 | \( 1 + (0.781 + 0.623i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.78969010453245905848096022475, −29.5634956743013136926591278557, −27.90048929405822770661229910655, −27.45399483323912077472799238136, −26.604309694064484255020775872867, −25.12592313341964556323220403189, −24.44693911864456173030348331077, −23.23811717470581743503890710940, −22.699869522486939373390494122103, −20.64220637410198339610649674529, −19.78781393825064781464993918608, −18.623647278255749721381931918233, −17.30641196028535679243521666918, −16.602420297174763054642563480949, −15.342124879814550961743754827489, −14.46756638739633890457510713476, −13.13389693814877893199516297356, −11.55710404910772801422816385165, −10.27662986874787205675652349200, −8.783825146700662472374161644639, −7.83810701058061526299049304319, −6.79279642265836277308239976997, −5.06203221410069970624827036409, −3.97417692785062555574987915012, −1.01007176919940394945893374550,
1.76278656609422774576018981287, 3.40409675488159037735244571356, 4.59807294545175725664841135302, 6.76140068455620403969994644849, 8.31223790796670270048965935818, 9.07675273154434479157061142269, 10.828795732215320060949493574060, 11.56555671697906283591914794865, 12.48102162141166837398924929792, 14.12462751628266548940937966380, 15.261273694926370803788994662109, 16.763947815572755154798282819542, 17.8899488047536465023697588316, 19.172370242360127086372394300185, 19.47596792079739411765948164293, 21.12881544146923272320641322746, 21.822304962139476013384465149564, 23.01061883213932677207471939630, 24.20963296890795461858664129229, 25.688847478806361998204580185076, 26.745531127389026743828022801549, 27.68149679130188100274771358881, 28.30279329174246934038692885713, 29.69723055814498570406504035614, 30.73514942026164304387000112153